sinx??1,x?031.(1)解 lim f(x)?lim??x?0x?0x?1,x?0即f(x)在x?0处的极限不存在，所以f(x)在x?0处不连续.
(2)解 ∵limf(x)?limx2sinx?0x?01?0?f(0)， x∴f(x)在x?0处连续.
(3)解 ∵limf(x)?limex?1，limf(x)?limx?0?x?0?x?0?sinx?1，f(0)?1，
x?0?x∴f(x)在x?0处连续.
32.不连续
33.解 limf(x)?lim(a?x2)?a，f(0)?1，
x?0?x?0?x?0?limf(x)?limln(b?x?x2)?lnb，
x?0?x?0?x?0?∵f(x)在x?0处连续，∴有limf(x)?limf(x)?f(0)， 即a?1，lnb?1，即a?1，b?e.
34.解 当x?0或x?0时，f(x)为初等函数，连续. 要使f(x)在(??,??)内连续，当且仅当f(x)在x?0处连续.
∵limf(x)?limxsinx?0?x?0?1?0，limf(x)?lim(a?x2)?a，f(0)?a.
x?0?x?0?x∴a?0.
x2?2x?512?2?1?5??4. 35.(1)解 lim22x?1x?11?12(2)?1?2 (3)
?ln1(?x)(4)解 ?ln1(?x)x在x?0处不连续，所以不能直接利用连续函数求极
x1限的法则.
令(1?x)?u，当x?0时，u?e. lnu在u?e处连续，所以
原式 =limlnu?lnlimu?lne?1.
u?eu?e1x?? 26
1??x?lim(1?x)另解：原式 =limln(1?x)?ln??x?0??lne?1. x?0??1x3 (5)? (6)
2sinsinxsinx?sinsinxx?1?1?1， ?limsinx36.证 ∵lim1x?0ln(1?x)x?0lneln(1?x)x∴sinsinx~ln(1?x).
37.(??,1)?(1,2)?(2,??)；
132
38.解 设f(x)?x2?2x?k，∵x?3时，x?3?0，∴f(x)为x?3时的无穷小，即有limf(x)?0.
x?3又∵f(x)是连续函数，∴有limf(x)?f(3)?0，即32?2?3?k?0，
x?3即k??3.
1?x?0，39.解 设f(x)?x2?ax?b，∵x?1时，∴有limf(x)?0. 又∵f(x)x?1连续，∴有limf(x)?f(1)?0，即1?a?b?0，即b??1?a (1).
x?1将(1)代入原式，得
x2?ax?1?a(x?1)(x?1)?a(x?1)lim?lim?lim[?(x?1)?a] x?1x?1x?11?x?(x?1)??2?a?5，即a??7，代入(1)，得b?6.
∴a??7，b?6.
40. 提示：利用零点定理 41. 略 42. 略
43.证 设f(x)?x?asinx?b，f(x)在[0,b?a]上连续，
f(0)??b?0，f(b?a)?b?a?asin(b?a)?b?a[1?sin(b?a)]?0，
(1)当f(b?a)?0，b?a即为所求；
(2)当f(b?a)?0，则在(0,b?a)内至少有一点?，使f(?)?0，?即为所求.
27
综上得证.
44. 略 45. 略
46.证 设g(x)?f(x)?x，g(x)在[a,b]上连续. ∵g(a)?f(a)?a?0，g(b)?f(b)?b?0，
∴根据零点定理，g(x)?0在(a,b)内至少有一个根，即，至少有一点??(a,b)，使得
g(?)?f(?)???0，即f(?)??.
47.(1)y??2x?3 (2)y??3cos(3x?1) 48.v(t)?dsdt?3t2，v(3)?27 49.y?6x?9 50.y?34(x?1) 51.(ln2,3) 52.切线方程为 x?2y?3?0；法线方程为 2x?y?1?0 53.解 在方程两边对x求导，得
3(5y?3)2?5?y??5(2x?1)4?2
将x?0，y??15代入上式，整理，得y??23.
所求切线方程为y?15?23x，或10x?15y?3?0； 法线方程为y?15??32x，或15x?10y?2?0. 54.a??2，b?4 55.a??1，b??1，c?1
56.证 设两曲线的交点为(x0,y0)，则有f(x0)?f(x0)sinx0，已知f(x)?0，∴有sinx0?1，从而有cosx0?0.
y?1(x0)?f?(x0)，y?2(x0)?f?(x0)sinx0?f(x0)cosx0?f?(x0)， 即，在交点(x0,y0)处两曲线的切线斜率相等，所以两曲线在交点处相切.
57.解 原式 =?f(x0??x)?f(x0)f(x0?(?3?x))??limx?0???x?f(x0)??3?x?(?3)??
?f?(x0)?3f?(x0)??3?3(?3)??12.
58.?1 59.(n?m)f?(a)
28
12sin2x?060.解 limf(x)?f(0)x?0x?limxx?0x?lim?sinx?x?0??x???1， ∴f?(0)?1.
?2? 当x?0时，f?(x)???1??sinx??xsin2x???????x?? ?(2sinxcosx)?x?sin2xxsin2x?sin2x2?xx2，
∴f?????0?14?2????2??????2. ?2??61.解
ddxf?g(x)??f??g(x)??g?(x). limg(x)?g(0)x2cos1?0x?limx?limxcos1?0，即g?(0)?0x?0x?0xx?0x. ∴
ddxf?g(x)?x?0?f??g(x)??g?(x)x?0?f??g(0)??g?(0)?f?(0)?0?0. 62.解 ∵limf(x)x?1x?1?2，∴当x?1时，f(x)与x?1为同阶无穷小，即有limx?1f(x)?0.
又∵f(x)在x?1处连续，∴有f(1)?limx?1f(x)?0.
从而 lf(x)x?i1mx?1?lx?i1mf(x)?f(1)x?1?2，即 f?(1)?2. 63.(1)?1?5x32xx (2)?1?1?45x3?16x2x??1?x?? (3)1?5x2 (4)
1?x(1?x2)1?x2 (5)
32(1?x)1?x (6)
12xlna (7)
1?1?112x??1?lnx?? (8)?x21?x2?x(1?x2) (9)x(1?x) (10)
ln5?5lntanx (11)?e?x1sinxcosx(cosx?sinx) (12)4?x2
(13)
12x?x2 (14)
11?x2 (15)21?x2
29
2arcsinx(16)?1xarccosx21?x2?(1?x2)1?x2 (17)4?x2
(18)exsinx(sinx?2cosx) (19)?2sinln(1?2x)1?2x (20)1xlnx
(21)21?x2 (22)0 (23)sinx?lntanx
(24)x2(cosx?xsinx)2 (25)sec2x2tanx2?csc2xx2cot2 (26)?4cos(2x)sin3(2x) (27)解 lny?a1lnx(?a1)?a2lnx(?a2)???anlnx(?an)，
1y?y??a1a2anx?a????， 1x?a2x?any??(x?a)a1(x?aa?a1??a2??an?12)2?(x?an)an??x?a????x?a?????x??1??2???an?. ? (28)解 lny?1xlnx，1y?y???1111x2lnx?x?x?x2(1?lnx)， y??xx?11x?2x2(1?lnx)?x(1?lnx).
(29)?x?1?x2?nnn?1n1?x2 (30)??x??2x?1??(2x?1)2
(31)解 lny?secxln1(?x2)，
1y?y??secxtanxln(1?x2)?secx2x1?x2， ∴y??(1?x2)secx??2x??secxtanxln(1?x2)?secx1?x2?? ?secx(1?x2)sexc??2x??tanxln1(?x2)?1?x2??. (32)解 lny?lnx?12?ln1(?x)?ln1(?x)?， 30
∴∴
一、极限
1.证明：lim
xx?0
x
不存在.
2.证明：当x?0时，4?x?2与9?x?3是同阶无穷小量. 3.证明：1?x?1～
x2(x?0).
4.当x?1时，两无穷小
1?x1?x和1?x中哪一个是高阶的？ 5.当x?1时，无穷小1?x和下列无穷小是否同阶？是否等价？ (1)1?3x; (2)2(1?x). 6.设当x?0时，1?cosx～asin2x2，求a的值. 7.求极限：
(1)xlimx3?2??2(x?2)2 (2)lim2x?3x?1x2?5x?4 (3)lim3?x?1?xx?1x2?1 (4)limsinxx??x (5)lim1x?0xsinx (6)limarctanxx??x
(7)limx?cosxx??2x?cosx
8.求极限：
(1)limx2?1x3x??2x?x?1 (2)lim?12 x??x4?5x?3 (3)lim?x3x2?x?????2x2?1?x2?2x?1??? (4)xlim3???3x3?1 (5)lim2n?3n?1?2?3???nn?n??2n?1?3n?1 (6)limn????n?2?2?? 1
1?1(7)lim2?14???12nn??1?111 3?9???3n(8)limn???1?2???n?1?2???(n?1)?
(9)lim?n???1?1?3?13?5?15?7???1?(2n?1)(2n?1)?? (10)limx2?2x?1x?1x3?x (11)lim?31?x?1??1?x3?1?x?? xn(12)lim?1(n?N) (13)lim4x3?2x2?xx?1x?1?x?03x2?2x (14)limx2?6x?8x?4x2?5x?4 (15)limx?0(1?cosx)cot2x
(16)lim3?x?1?xx?1x2?1 (17)limx2x?01?1?x2 (18)lim1?x?1?xx?0x (19)2xlim????x?x?x?
(20)xlim???x?x2?1?x? (21)limx???x2?1?x2?1?(22)2xlim????x2?x?x?x?3 (23)limx?1x?1x?1
(24)lim1?x?3x??82?3x (25)lim1?x?1x?031?x?1
*9.若lim?x????x2?1?x?1?ax?b?????0，求a，b的值. 10.求下列极限：
x2?sin1(1)limxx?0sinx (2)limx2x?0
sin2??x??3??(3)limtan2xx?0sin5x (4)limx?0x?cot3x
(5)lim1?cosxx?0sin2x (6)
lim2arcsinxx?03x
2
(7)limxx?0?1?cosx (8)limn??2nsinx 2n(9)limtanx?sinxx?0sin3x (10)lim?x????3x2?5?5x?3?sin2?x??? x(11)lim1?x?1?xx?0sinx (12)lim?x???1??1?x??
21(13)lim(1?x)x 2xx?0 (14)limx?0(1?3x)
xx(15)xlim?1??????1?x?? (16)lim?x?x????1?x??
2xx?(17)lim?2??2?21x????1?x?? (18)limx????1?x??
2x(19)lim?xx?0?2?x??2?? (20)lim?x?1?x????x?1?? x?1x?1(21)lim?x?a?x?? (22)lim?2x?3????x?a?x????2x?1??
?x2?x(23)limx??????x2?1?? (24)lim(1?cosx)3secxx?? 2(25)lim?1?sinx?1xx?0 (26)limx?0(1?3tan2x)cot2x
x11.已知lim??x?a???a???4，求常数ax??x.
二、连续函数
12.设函数?(t)?t3?1，求?(t2)，??(t)?2.
13.设f(x)?1?x1?x，(x??1)，求f(?x)，f(1?x)，f??1??x??. 14.设f(x)?xx?1，求f(x?1)，f??x??x?1??. ?1,x?15.设f(x)??0?0,x?0，求f(x?1),f(x2?1).
??1,x?03
?1,16.设f(x)???0,x?1，求f?f(x)?. x.?1?x2,0?x?117.设?(x?1)??，求?(x).
2x,1?x?2?18.设函数z?x?y?f(x?y)，当y?0时，z?x2，求f(x)及z.
?1,0?x?119.设f(x)?? ，求函数f(x?3)的定义域.
??2,1?x?220.设f(x)为定义在(?a,a)上的奇函数，且f(x)在[0,a)上单调减少. 试证明：
f(x)在(?a,0]上也单调减少.
21.设函数f(x)在(??,??)内单调增加，且对一切x有f(x)?g(x). 证明：
f?f(x)??g?g(x)?.
22.证明任一定义在区间(?a,a)(a?0)上的函数可表示为一个奇函数与一个偶函数之和.
23.求下列各函数的定义域： (1)y?(3)y?(5)y?1x?1?x?2y?arcsin (2)
21?x2lg(3?x)5x?x2
(4)y?lg4x?11?x?2 (6)y?16?x2?sinx
lg(1?x)21?x(7)y??ln(2x2?x) x24.求函数y?arccos2x的定义域与值域. 21?x25.求下列函数的反函数：
2x10x?10?x?1 (1)y?x (2)y?x?x10?102?1ex?1(3)y?x (4)y?32x?5
e?1
4
(5)y?1?lg(x?2) (6)y?1?1?4x1?1?4x
26.已知f(x)?sinx，f??(x)??1?x2，求?(x)及其定义域. 27.设函数f(x)的定义域为??1,0?，求下列各函数的定义域：
(1)f(x3) (2)f(sin2x) (3)f(x?a)?f(x?a) (a?0)
128.求函数y??x2?x当x?1，?x?0.5时的增量.
229.求函数y?1?x当x?3，?x??0.2时的增量. 30.若f(x)?cos2x，求limf(x??x)?f(x).
?x?0?x31.下列函数f(x)在x?0处是否连续？为什么？
?sinx1?2xsin,x?0,x?0??xx??(1)f(x)?? (2)f(x)??
?0,?1,x?0x?0?????x?e,x?0?(3)f(x)??
?sinx,x?0??x???1,???x?0?32.讨论函数f(x)?? 在x?0处的连续性.
?21xsin?x,0?x????sinx?a?x2,x?0?x?0，已知f(x)在x?0处连续，试确定a，b的值. 33.设f(x)??1,?ln(b?x?x2),x?0?1?xsin,x?0?x?34.设f(x)??，要使f(x)在(??,??)内连续，应当怎样选择a？
?a?x2,x?0??
5
习题答案与部分题解
1.证 ∵limxx?limx?xx?lim?lim(?1)??1， ?lim1?1，limx?0?xx?0?xx?0?x?0?xx?0?xxx?0?即 limxxx?0??limxxx?0?， ∴limx?0不存在.
2.证 ∵limx?04?x?29?x?3?3， 2?limx?0(4?x?4)(9?x?3)(9?x?9)(4?x?2)
?limx?09?x?34?x?2∴4?x?2与9?x?3是同阶无穷小量.
3.提示：只要证明limx?01?x?1?1. x24.为等价无穷小 5.(1)为同阶无穷小 (2)为等价无穷小 6.a?2
(x?2)20??0， 根据无穷大与无穷小的关系，∴原式 =?. 7.(1)解 ∵lim3x??2x?2?6(2)?（根据无穷大与无穷小的关系） (3)?
1(4)解 当x??时，?0，sinx?1，根据无穷小与有界量的乘积仍为无穷小，
x∴原式 = 0 .
(5)0（根据无穷小与有界量的乘积仍为无穷小） (6)0
1cosx1x(7)解 原式 =lim?.
x??122?cosxx1?x2?118. (1)解 lim2?lim?. （无穷小量分出法）
x??2x?x?1x??1122??2xx (2)0 （利用无穷小量分出法）
21
1?1x2??x3x2?x3(2x?1)?x2(2x2(3)解 limx????2x2?1?2x?1????lim?1)x??(2x2?1)(2x?1) 1?1?limx3?x2x??(2x2?1)(2x?1)?limxx????1.
??2?1??1?4x2????2?x??1?3(4)解 x2?3xlim?x3?1?x2??3xlim????1. 31?1x3(5)13 (6)?12 (7)423 (8)2 (9)lim?1n????1?3?13?5?15?7???1?(2n?1)(2n?1)?? ?lim1?n??2?????1?1?3?????1?3?1?5?????1?5?1?7???????1?2n?1?1?2n?1????? ?lim1?1n??2??1??2n?1???12. 注：令
1AB(2n?1)(2n?1)?2n?1?2n?1，得
1?A(2n?1)?B(2n?1)?(2A?2B)n?A?B.
比较等式两边n的同次幂的系数，得
??2A?2B?0??A?12?A?B?1 ，解得 ?B??12. 于是
1(2n?1)(2n?1)?1?2?1?2n?1?1?2n?1??. （此法称为待定系数法）
(10)0 (11)1 (12)n (13)12 (14)213 (15)2 (16)解 lim3?x?1?x(3?x)?(1?x)x?1x2?1?limx?1(x2?1)(3?x?1?x) ?lim2(1?x)?2x?1(x2?1)(3?x?1?x)?limx?1(x?1)(3?x?1?x)??24.(17) ?2 (18) 1 (19)
112 (20) 2 (21) 0 (22)1
22
(23)
23 (24) ?2 (25)解 lim1?x?1[(1?x)?1](3(1?x)2?31?x?1)x?031?x?1?limx?0[(1?x)?1](1?x?1)
3?lim(1?x)2?31?x?1x?01?x?1?32. 9.解 原式 =limx2?1?(ax?b)(x?1)x??x?1
?lim(1?a)x2?(a?b)x?1?bx??x?1?0 要使上等式成立，当且仅当 ??1?a?0，即 a?1，b??1.
?a?b?0x2sin1110.(1)解 lxxsinx?i0msinx?lx?i0mxsinx?0. x(2)9 (3)25 (4)13 (5)12 (6)23
x(7)解 xxxlim?0?1?cosx?xlim?0??lim2?2?2.
2sinxx?0?x2sin2(8)x (9)12
2?(10)解 lim?3x?5?3?52?x?????5x?3?sin2?x????lim?x2sinx?6x????3?2?2???5. ?5?xx??(11)解 lim1?x?1?x(1?x)?(1x?0sinx?lim??x)x?0sinx1?x?1?x? ?lim2x?0sinxx??1.
1?x?1?x?x?x??1(12)解 lim?x????1?1?x???1???limx?????1?????x???e?1.
?
23
3(13)e?2 (14)e?2 (15)1
x(16)解 l?x??1?xx?i?m??1?x???lx?i?m??1?1?x??
x?1???(1?x)x)?(1?x)?lim??(1?1?1xx??????1?1?1?x????lim??1???e?1.
???x?????1?1?????x???(17)e4 (18)e?1 (19)e?1 (20)e?2 (21)e2a (22)e
xx(23)解 lim?x????x2???x2?1???lim?x????1?1?x2?1?? x12?1?lim???x?x2?12?1?1?1?2??lim?x?x?1x?????x?1???x?????1?1?x2?1???x??????e0?1.?(24)e3 (25)e (26)e3 11. ln2
12. t6?1；t6?2t3?1
13.
1?x1?x,(x?1)；x2?x,(x?2)；x?1x?1,(x??1且x?0) 14.
x?1x?2,(x?2)；x,(x?1) 15.解
?1,x?1?0f(x?1)???0,x?1?0
??1,x?1?0???1,x?1； ?0,x?1?1,x2?1?0f(x2?1)???0,x2?1?0
??1,x2?1?0 ???1,x??1?0,x??1 .
16.1
24
17.解 令x?1?t，则x?t?1，
?(t)???(t?1)2,0?t?1?1 ?2(t?1),1?t?1?2 ???(t?1)2,1?t?22(t?1),2?t?3 ，
?x)???(x?1)2∴?(,1?x?22(x?1),2?x?3
?18.f(x)?x2?x；z?(x?y)2?2y
19. ??3,?1? 20. 略 21. 略 22. 略 23. (1)??2，?1????1，1???1，??? (2)??1，3?
(3)???，?1???1，3? (4)?1，4? (5)??2,0???0,1?
(6)??4,?????0,?? (7)??1,0????1??2,1??
24.定义域为???，???；值域为?0，??.
25. (1)y?logx21?x，(0?x?1) (2)y?1x2lgx?2，(x?0或x?2）(3)y?ln1?x1?x，(?1?x?1) (4)y?12?log3x?5?，(x?0)
(5)y?10x?1?2，(??,??) (6)y??x(1?x)2,(?1?x?1)
26.?(x)?arcsin(1?x2)；??2,2?
27. (1)??1,0?
(2)??????k?1?2???,(k?1)????，k?Z 或??????k?1?2???,k????，k?Z
(3)当0?a?12时，定义域为?a?1,?a?；当a?12时，定义域为?. 28.?1 29.?0.0506 30.?2sin2x
25
(12)limln(1?2x)tanx?x (13)lim
x?0x?0sin3xx?sinxxn(14)limax （n为正整数，a?0）
x???eex?e?x?2xln(1?ex)(15)lim (16)lim
x?0x?sinx(17)limtanxx??tan3x 2(19)limln(1?x2)x?0secx?cosx ln??1?1??(21)xlim?x????arccotx (23)limcosx?cos3xx?0x2 (25)limx?1(1?x)tan?x2 (27)xlim3?0?xlnx (29)lim?x?0?1?x?1?ex?1?? (31)lim?x?0??1?ln(1?x)?1?x??? (33)lim??1x???x????x???e?1??? ???(35)lim?1x???(2?x)ex?x?? ??1(37)lim1?xx?1x *(39)lim??cosx??2?x x?2?0x???1?x2 (18)lntan7xxlim?0?lntan2x
(20)limx?arcsinxx?0sin3x
(22)limx?arctanxx?0x?arcsinx
(24)cosx?ln(x?axlim)?a?ln(ex?ea) 1(26)limx?0x2ex2
(28)lim?0?xmxlnx(m?0)(30)lim?xx?1??x?1?1?lnx?? (32)lim?11?x?0??x2?xtanx?? (34)limx???x?????2?arctanx?? 1(36)limx?0(1?sinx)x
tanx(38)xlim??0??1??x??
1?x2 *(40)lim?x?x?1???x?1?2??
11
1*(41)limxx?0?ln(ex?1) *(42)lim?1?xe2x?01x1?cosx?
???*(43)lim??arctanx?x???2??1lnx11?1?ax?a2x???anx *(44)lim?1x???n??? ???nx80.证明函数y?2x?x2在区间(0,1)上单调增加，而在区间(1,2)上单调减少.
81.求下列函数的单调区间：
(1)y?2x3?12x2?18x?5 (2)y?x4?2x2?5 (3)y?(x?2)5(2x?1)4 (4)y?x?ln(1?x) 82.证明不等式：
(1)2x?3?1x(x?1) (2)x?ln(1?x)(x?0)
(3)ln(1?x)?arctgx1?x(x?0) (4)x?x22?ln(1?x)?x (x?0) (5)e?x?sinx?1?x22 (0?x?1)
(6)arctanx?x (x?0)，arctanx?x (x?0). 83.求下列函数的极值：
(1)y?2x3?3x2 (2)y?2x2?x4
(3)y?2x3?6x2?18x?7 (4)y?1x4?133x?x24 (5)y?(x?1)2(x?2)3 (6)y?xlnx
(7)y?(x?1)?3x2 (8)y?2x?3?3x2
(9)y?3x2?4x?4x2?x?1 (10)y?2?x?x2
12
84.求下列函数在所给区间的最大值与最小值： (1)y?x5?5x4?5x3?1,[?1,2]
1?x?x2,[0,1] (2)y?x?2x,[0,4] (3)y?21?x?xx2?1?85.求函数y?的单调区间，并求该函数在区间??,1?上的最大值与最小
1?x?2?值.
86.试证方程x3?x?1?0只有一个正实根. 87.讨论方程xe?x?a(a?0)有几个实根.
88.欲做一个底为正方形，容积为108立方米的长方体开口容器，怎样做法所用材料最省？
89.欲用围墙围成面积为216平方米的一块矩形土地，并在正中用一堵墙将其隔成两块，问这块土地的长和宽选取多大的尺寸，才能使所用建筑材料最省？
90.欲做一个容积为300立方米的无盖圆柱形蓄水池，已知池底单位造价为周围单位造价的2倍. 问蓄水池的尺寸应怎样设计才能使总造价最低？
五、不定积分
91.求下列不定积分：
(1?x)21?x?(1)??2?2sinx??x?1?dx (2)?3dx
xx??(3)?(5)?x2?22x?2x?21?x21??dx (4)??1?2?xxdx
?x?1?2x2dx (6)?2dx 24x(1?x)1?xx42?3x?5?2xdx (8)?dx (7)?2x1?x3xx??(9)?bedx (10)??sin?cos?dx
22??xbx24sin3x?12x2sindx dx(11)? (12)?22sinx
13
cos2x1?cos2xdx (14)?dx (13)?cosx?sinx1?cos2x92.求下列不定积分：
(1)?f?(x)dx (2)?f?(2x)dx (3)??f(x)?xf?(x)?dx 93.设f?sin2x?cos2x???x?1?，求f(x).
3，并当x?1时，这个函数值等于?，
21?x294.已知一个函数的导函数为f(x)?求这个函数F(x).
195.已知曲线y?f(x)上任一点的切线的斜率为ax2?3x?6，且x??1时，y?是极大值，求f(x)和f(x)的极小值.
11296.已知f(x)的图形过点(0,3)，f?(x)的图形是过点(1,0)且不平行于坐标轴的直线，2是f(x)的极值，求f(x).
97.求下列不定积分：
(1)?(a?bx)kdx (b?0) (2)?(3)?3(5)?dx3?2xdx (4)?3dx
(1?2x)2dxx?1?x?1?2x2?1
x31?x2dx (6)?xe
dx
(7)?cos(???x)dx(9)?22 (8)?cosxsinxdx
cosxxdx (10)?1dx
1?cosx(11)?cosx?sinx2?lnxdx (12)?dx
1?2sinxcosxx?1?ln?1??(13)4?x?dxsin (14)?xdx ?x(x?1)3(15)?cosxdx
35(16)?sinxcosxdx
14
3cosx(17)?tan5xdx (18)?dx 4sinxcos3x4dx (20)?secxdx (19)?sinx244(21)?tanxdx (22)?(tanx?tanx)dx
2(23)???secx??1?tanx??dx (25)?xdx2?x4 ?dx(27)4?x2arcsinx 2(29)?x?1x2?x?1dx (31)?x5?3(1?x3)2dx (33)?14?9x2dx (35)?11?exdx 98.求下列不定积分：
(1)?x2(1?x)100dx (3)?dx1?1?x (5)?dx (1?x2)32(7)?dxx2?a2 (9)?14?9x2dx (24)?12?3x2dx
(26)?1x2?8x?25dx (28)?dxx?x2 (30)?x?5x2?6x?13dx lntanx(32)?2sinxdx
(34)?11?exdx (36)?ex?1ex?1dx (2)?1x(1?3x)dx (4)?dxex?1
(6)?x2dxa2?x2
(8)?13dx
(x2?a2)2 (10)?1xx2?1dx
15
(11)?dxx2x2?9 (12)?13x2?2dx
(13)2?x?a2xdx
99.求下列不定积分： (1)?te?2tdt (2)?x2exdx
(3)?xcosx2dx
(4)?xcos2xdx (5)?xsinxcosxdx (6)?xtan2xdx (7)
?1sin2xcosxdx (8)?sec3xdx (9)?ln2xdx (10)?x2arctanxdx
(11)?x21?x2arctanxdx (12)?e?2xsinx2dx (13)?sinlnxdx (14)?(arcsinx)2dx (15)?xln(1?x2)dx (16)?x2ln(1?x)dx
2(17)???lnx?lnsinx?x??dx (18)?sin2xdx (19)?arcsinxx2 (20)?xexdxex?1dx
(21)?exx(1?xlnx)dx (22)?e?xarctanexdx
*(23)?xexsinxdx 100.设f(x)的原函数为
sinxx，求?xf?(x)dx. 101.设f?(ex)?1?x，求f(x). 102.求下列不定积分：
(1)?x?1(x?1)(x?2)dx (2)?x3?14x3?xdx 16
(3)?x33?xdx *(4)?x5?x4?8x3?xdx *(5)?1x2?1(x2?1)(x2?x?1)dx *(6)?(x?1)2(x?1)dx
六、定积分
103.求下列函数的导数： (1)f(x)??xx301?tdt (2)f(x)??sint2dt
0x2?1(3)f(x)??101?t4dt
(4)f(x)??te?txdt
(5)f(x)??x2ttx3edt
(6)f(x)?x2?2x0e2dt
(7)f(x)??2extanln(t2?1)dt (8)f(x)??cosxcostsinx1?t2dt *104.设f(x)是连续函数，且?x3?10f(t)dt?x，求f(1).
?x??0tf(t)dt?105.设F(x)??x2,x?0? ，其中f(x)有连续的导数且f(0)?0. 研究：
??0,x?0??(1)F(x)在x?0处的连续性；(2)F(x)在x?0处的可导性.
*106.试求由?yetdt??x00costdt?0所确定的隐函数对于x的导数y?.
*107.设x?y2??y?x20costdt，求
dydx. 108.求下列极限：
xx(1)lim?0cos2tdtx?0x (2)lim?0arctantdtx?0x2
109.判断函数f(x)??x3t?10t2?t?1dt在区间?0,1?上的单调性. 110.求函数f(x)??x0te?tdt的极值.
111.求函数f(x)??x0t(t?4)dt在??1,5?上的最大值与最小值.
17
*112.设函数f(x)在(0,??)内可微，且f(x)?1?1x?x1f(t)dt，试求f(x). 113.求下列定积分： (1)
???620(2cos2??1)d? (2)?0cosxsin2xdx
(3)??(1?sin3?)d? (4)
2sin10?1?xdx 2?x(5)?3dx1x?x2 (6)?10?ex?1?4exdx 1e(7)?dx (8)?1?lnx 0ex?e?x 1xdx(9)?e3dx a2rdr1x1?lnx (10)?03a2?r2(a?0)
(11)?0dx ?2x2?2x?2114.设f(x)在??b,b?上连续，试证
?bb?bf(x)dx???bf(?x)dx.
115.证明?a2a?af(x)dx?2?0f(x2)dx，其中f(x)为连续函数.
116.证明?1dx1x1?x2??xdx11?x2(x?0).
117.证明?10xm(1?x)ndx??10xn(1?x)mdx.
118.求下列定积分：
(1)?1(1?x2)3dx 4
0 (2)??1xxdx(3)?11?x21x2dx (4)?a20x2a2?x2dx (5)?1x2 0(1?x2)2dx (6)?2x2?11xdx 3(7)?4dx (8)?1x201?x 01?xdx
(9)?5x?11xdx (10)?1xdx?15?4x 18
(11)?ln20e?1dx (12)?x2ln2?20xsin(2x2)dx 21?sin(x)119.已知?a?dx?，求a.
6ex?11120.求下列定积分：
(1)tedt (2)?ecosxdx?x1?t220?20
(3)?ln(x?1)dx (4)
010e?1?320earccosxdx
(5)?xarctanxdx (6)?sin(lnx)dx
1xsinxdx 30cosxe?
(9)?x2cos2xdx (10)?1lnxdx(7)?xcosxdx (8)?02?2?40e121.已知常数b?0，且?lnxdx?1，求b的值.
1b?1?1?x,x?02?*122.设f(x)??，求?f(x?1)dx.
0?1?1?ex,x?0?123.求下列各题中平面图形的面积： (1)曲线y?a?x2(a?0)与x轴所围成的图形. (2)曲线y?x2?3在区间[0，1]上的曲边梯形. (3)曲线y?x2与y?2?x2所围成的图形. (4)曲线y?x3与直线x?0、y?1所围成的图形.
?(5)在区间?0,???上，曲线y?sinx与直线x?0,y?1所围成的图形. ?2?(6)曲线y?1与直线y?x,x?2所围成的图形. x(7)曲线y?x2?8与直线2x?y?8?0,y??4所围成的图形.
124.求由抛物线y?x2?4x?5，横轴及直线x?3,x?5所围成的图形的面积. 125.求由曲线y?xe?x，横轴及直线x?0,x?1所围成的图形的面积.
19
2126.求由曲线y?lnx，纵轴与直线y?lna，y?lnb(b>a>0)所围成的图形的面积. 127.求由抛物线y?3?2x?x2与横轴所围成的图形的面积. 128.抛物线y?12x分割圆x2?y2?8成两部分，分别求出这两部分的面积. 2129.求下列平面图形分别绕x轴、y轴旋转产生的立体的体积： (1)曲线y?x与直线x?1,x?4,y?0所围成的图形.
?(2)在区间?0,????x?,y?0所围成的图形. 上，曲线与直线y?sinx?22?(3)曲线y?x3与直线x?2,y?0所围成的图形. (4)曲线x2?y2?1与y2?3x所围成的两个图形中较小的一块. 2130.求曲线xy?a(a?0)与直线x?a，x?2a及y?0所围成的图形绕x轴旋转所成旋转体的体积.
131.设平面图形由y?ex，y?e，x?0所围成，(1)求此平面图形的面积；(2)求此平面图形绕x轴旋转所形成的旋转体的体积.
132.证明圆锥体的体积为其底面积与高的乘积的三分之一. 133.求下列广义积分： (1)?edx (2)
0???x???1???xdx(3)xedx ?0
x(4)
???e????arctanx?xdx(5) (6)esinxdxdx ?2?02 1xx(lnx)(7)??1(9)?021dx1?x2 (8)?12dx(x?1)a(a?0)
1 22dxdx(10)?0xlnxdx (11)?02 2 x?4x?3(1?x)134.计算y?e?x与直线y?0之间位于第一象限内的平面图形绕x轴旋转产生的旋转体的体积.
20
111?11?111?x?x2y?y??x?2???1?x?1?x???x?1?x2?x(1?x)(1?x)， y??1?x1?x?1?x?x2∴(1?x)(1?x).
(33)x?33x?a?2x?b?11?x?3x?a?1?2x?b??
(34)
x233?x??21x?1?x(3?x)2?x?1?x?9?3(9?x2)?? 64.(1)解 在方程两边对x求导，得
2x?2y?y??y?xy??0，
整理，得 y??y?2x2y?x. (2)y??ayy?ax (3)y??yy?1 (4)y??ey1?xey (5)y??tan2x 65.解 在方程两边对x求导，得
1xy??yx2?y2?12x2?y2?(2x?2y?y?)?12?1???y?x2， ?x??x?y?y?xy??yx2?y2?x2?y2， ∴y??x?yx?y. 66.y???yx 67.(1) 解 y??f?(ex)exef(x)?f(ex)ef(x)f?(x) ?ef(x)?ex?f?(ex)?f(ex)?f?(x)?. (2)?1xx2?1?f????arcsin1??? (3)(ex?exe?1)?f?(ex?xex)
(4)sin2x??f?(sin2x)?f?(cos2x)? 68.?1(1?x)2 31
69.(1)2sin(5x?3)?20xcos(5x?3)?25x2sin(5x?3)
(2)2?2x2(1?x2)2 (3)1x (4)2arctanx?2x1?x2
(5)2x(2x2?3)ex2 (6)?2cos2xlnx?21xsin2x?x2cos2x 70.(1)解 y??f?(ex)ex?exf?(ex)，
y???exf?(ex)?exf??(ex)ex?exf?(ex)?e2xf??(ex).
(2)解 y??12f(x)f?(x)，
f??(x)f(x)?f?(x)f?(x)2f(x)f(x)?2f(x)f??(x)??f?(x)?2y???12.
4?f(x)?32(3)ef(x)?f??(x)?2?f?(x)?2?f(x)f??(x)?f(x)?f?(x)?2? (4)?1f(x2)?4f?(x2)?2(lnx)f?(x2)?4x2(lnx)f??(x2x2) (5)2f?(x2)?4x2f??(x2)?f(x)f??(x)??f?(x)?2?f(x)?2
71.解 在方程两边对x求导，得
eyy??y?xy??0 (1)
再在(1)两边对x求导，得
ey(y?)2?eyy???y??y??xy???0 ey(y?)2?(ey?x)y???2y??0 (2)
当x?0时，由原方程解得y?1. 将x?0，y?1代入(1)，得y?(0)??1e.
将x?0，y?1，y???1e代入(2)，得y??(0)?1e2.
72.?0.02
73.(1)1?x2?xdx(1?x2)2dx (2)?e(cosx?sinx)dx (3)2x?x2 (4)xdx2x??x1?21?1?x2 (5)2(e?e2x)dx (6)earccosx???arccos?dx?xx2?1x??
32
(7)(sinx)cosx???cos2xsinx?sinx?lnsinx????dx ?74.解 由题设，x由x0?9变到x0??x?8.99，得?x??0.01.
y??ex1x?32x，y??1x?9???2xe?1?x?9?6e.
f?(x?13∴所求微分dy?3e0)?x6e(?0.01)??600.
75.
33t?1?sin?2ln(3t?1)?dt 76.证 设f(t)?lnt，t??x,1?x?，(x?0). ∵f(t)在?x,1?x?上满足拉格朗日中值定理的条件. ∴在(x,1?x)内至少存在一点?，使得
f(1?x)?f(x)?f?(?)[(1?x)?x]，
即有 ln(1?x)?lnx?1?. ∵x???1?x，∴
11?x?1??1x，从而有 11?x?ln(1?x)?lnx?1x. 77.提示：设f(x)?xp，x?[a,b]. 利用拉格朗日中值定理证.
78.证 由题设，F(x)在[0,1]内连续，可导，且有 F(0)?0，F(1)?f(1)?0，
根据罗尔定理，在(0,1)内至少有一点?1，使得F?(?1)?0.
又 F?(x)?2xf(x)?x2f?(x)，它在[0,?1]内连续，可导，且有
F?(0)?F?(?1)?0，根据罗尔定理，在(0,?1)?(0,1)内至少有一点?，使得F??(?)?0.
(2)1?a??a?179.(1)? n (3)a (4)3 (5)lna (6)2
33
(7)?1 (8)?2 (9)? (10)0 (11)?12 (12)23
(13)2 (14)0 (15)2
exln(1?ex)??1?ex1?1(16)解?ex1?x22xlim???1?x2?xlim???xxlim???x(1?ex)?xlimx????1. 1?x21?1ex(17)3
?7(18)解 lntan7x?xl?i0m?lntan2x?xl?i0mln7x?ln2x?xl?i0m7x?2?xl?i0m?1?1.
2x（当x?0时，tan7x~7x，tan2x~2x）
2x2x20(19)解 limln(1?x)02sinxx?0secx?cosx?lim1?xx?0secxtanx?sinx?limx?0 (1?x2)??1??cos2x?1???2?11?2?1. (20)解 limx?arcsinxx?0sin3x?limx?arcsixnx?0x3 (x?0,sinx~x) 01?1 ?0lim1?x22(1?x2)?1x?03x2?lim1?x?1x?03x21?x2?limx?03x21?x2(1?x2?1) ?l?11x?i0m31?x2(1?x2?1)??6.
(21)1 (22)?2 (23)4 (24)解 ∵xlim?a?cosx?cosa，
?1ln(x?a)?ex?ea00exlim?a?ln(ex?ea)?xlimx?ax?a?ex?xlim?a?ex(x?a)?xlim?a?ex(x?a)?ex ex?eaea?0?ea?1. 34
∴ 原式 =limcosxln(x?a)x?a??xlim?a?ln(ex?ea)?cosa?1?cosa.
(25)
2? (26)? (27)0 (28)0 (29)12 (30)12 (31)12 (32)13 (33)1 (34)1
(35)解 lim?1x??11x??x??x???(2?x)e?x???lim?x????2e?x??e?1?????， ??1∵limx??2ex?2?e0?2，
11ex??1?? limx??x?1??e?1??limx?10?0lim?x??1x?e?limex?0???x??1x???e?1， x??1?x???x??∴原式 =2?1?3.
1(36)解 令y?(1?sinx)x，则lny?ln(1?sinx)x， 0limx?0lny?limln(1?sinx)0cosxx?0x?limx?01?sinx?1， ∴limx?0y?e，即原式 =e.
3(37)e?1 (38)1 (39)1 (40)e2 (41)e (42)e21(43)解 令y?????lnxln????arctanx??2?arctanx??，则lny??2??lnx， ln????arctanx???xlim2???lny?xlim?x????lnx???1?x2xlim???? 2?arctanx1?x2?x?2x0?10(1?x2lim)2??lim1?x2?limx2?1x????1x???1?x2x???1??1， 1?x2x2?1
35
∴xlim???y?e?1，即，原式 =e?1.
?111?ax?a?nx(44)解 令y?2x???a?1nx????n?，则 ?lny?nx???ln?111??a???1x?a2x???anx??lnn??，
ln?11?a1?ax?a?12x??nx??limlny?nlim??lnnx??x??1 x1?1110111??a1xlna??1?1?a2xlna2???anxlnan?????0a1x?a2x???anx??x2???1x2a1?a11?nlim1xlna12xlna2???anxlnanx??a111
1x?a2x???anx?n?lna1?lna2???lnann?ln(a1a2?an)，
n∴limx??y?a1a2???an，即原式 = a1a2?an??ai.
i?180. 略
81.(1)解 y??6x2?24x?18?6(x?1)(x?3)， 令y??0，得驻点：x1?1，x2?3.
x (??,1) (1,3) (3,??) y? ＋ － ＋ y ↗ ↘ ↗ ∴单调增加区间：(??,1)?(3,??)；单调减少区间：(1,3).
(2)单调增加区间：(?1,0)?(1,??)；
36
单调减少区间：(??,?1)?(0,1)
(3)单调增加区间：?????,?1??11?2?????18,????；
单调减少区间：??111???2,18??
(4)解 函数的定义域为(?1,??).
y??1?11?x?x1?x，令y??0，得驻点：x?0. 当?1?x?0，y??0；当x?0，y??0.
(0,??)；单调减少区间为(?1,0).
82.(1)证 设f(x)?2x?3?1x，f(x)在[1,??)上连续， 3f?(x)?1x?1x2?1x2?x2?0，(x?1) f(x)在(1,??)上单调增加.
又 ∵f(1)?0，∴当x?1时，有f(x)?f(1)?0. 即
2x?3?1x，(x?1). (3)证 设f(x)?(1?x)ln(1?x)?arctanx，
f?(x)?ln(1?x)?1?11?x2?0，(x?0). f(x)在(0,??)上单调增加. 又 ∵f(x)在[0,??)上连续，且f(0)?0，x?0时，有f(x)?f(0)?0，即
(1?x)ln(1?x)?arctanx?0， ln1(?x)?arctaxn1?x，(x?0).
83.(1)解 函数的定义域为(??,??).
y??6x2?6x?6x(x?1)，令y??0，得驻点：x1?0，x2?1.
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∴函数的单调增加区间为∴∴ ∴当 x y? (??,0) 0 (0,1) 1 (1,??) ＋ ↗ 极 大 － ↘ 极 小 ＋ ↗ y ∴x1?0为极大点，函数的极大值为y(0)?0；x2?1为极小点，函数的极小值为
y(1)??1.
(2)极大值：y(?1)?1，y(1)?1；极小值：y(0)?0 (3)极大值：y(?1)?17；极小值：y(3)??47 (4)极大值：y(0)?0；极小值：y(?1)??512，y(2)??83 (5)极大值：y(1)?0；极小值：y??7?108?5????3125
(6)极小值：y(e)?e
(7)解 函数的定义域为(??,??).
y??3x2?(x?1)?23?15x?223x?3?3x，令y??0，得驻点：x1?5，
函数的不可导点：x2?0.
x (??,0) 0 ??2?2?0,5?? 5 ??2??5,???? y? ＋ － ＋ y ↗ 极 大 ↘ 极 小 ↗ 3∴x2?2?341?5为极小点，函数的极小值为y??5????5?25；x2?0为极大点，函数的极大值为y(0)?0.
(8)极大值：y(?1)?1；极小值：y(0)?0 (9)极大值：y(0)?4；极小值：y(?2)?83
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?1?3(10)极大值：y???
?2?284.(1)解 y??5x4?20x3?15x2?5x2(x?1)(x?3). 令y??0，得驻点：x1?0，x2?1，x3?3（舍去）. ∵y(?1)??10，y(0)?1，y(1)?2，y(2)??7， ∴函数的最大值为y?2；最小值为y??10.
(2)最大值：y?8；最小值：y?0 (3)最大值：y?1；最小值：y?3 585.单调增加区间为(??,?2)?(0,??)；
单调减少区间为(?2,?1)?(?1,0).
1?1?函数在??,1?上的最大值为，最小值为0.
2?2?86.证 设f(x)?x3?x?1，∵f?(x)?3x2?1?0，∴f(x)在(??,??)上单调增加. 又∵f(x)在[0,1]上连续，且
f(0)??1?0， f(1)?1?0，
∴由根的存在定理及f(x)的单调性可知，方程f(x)?0在(0,1)内有且仅有一个实根，即方程只有一个正实根.
87.当a?e?1时，方程无实根；当a?e?1时，方程有一个实根；
当a?e?1时，方程有两个实根.
88.解 设容器的底边长为x，高为y，则其容积为
108. x2432容器的表面积为 A?x2?4xy?x2?，(x?0).
x432A??2x?2，令A??0，得驻点：x?6.
xx2y?108，即y? 由问题的性质知，容器的最小表面积一定存在. 现在只求得唯一驻点，故当底
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边长x?6米时，容器表面积最小. 当x?6时，y?3. 即容器的底边长6米，高3米时，所用材料最省.
当底边长x?6米，高y?3时，所用材料最省.
89.当土地的长x?18米，宽y?12米时，所用建筑材料最省.
390.当池底半径r?150?米，高为底半径的2倍时，总造价最低. 2x391.(1)ln2?2cosx?lnx?23x2?x?C
258(2)32x3?65x3?38x3?C (3)12x2?2x?C
71(4)47x4?4x?4?C (5)arcsinx?C (6)arctanx?1x?C (7)?x?13x3?arctanx?C (8)2x?5?ln2?ln3?2?x?3???C (9)
bxebxb?lnb?C (10)x?cosx?C (11)?4cosx?cotx?C (12)x?sinx?C (13)sinx?cosx?C
(14)12tanx?12x?C
92.(1)解 ?f?(x)dx??df(x)?f(x)?C. (2)解 ?f?(2x)dx?12?df(2x)?12f(2x)?C. (3)xf(x)?C 93.f(x)?x?12x2?C 94.F(x)?arcsinx?? 95.f(x)?x3?32x2?6x?2，极小值为f(2)??8 96.解 由题意，设f?(x)?k(x?1)，k?0. 从而有
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f(x)??f?(x)dx??k(x?1)dx?k???x2?2?x????C.
?∵f(x)的图形过点(0,3)，∴有f(0)?C?3.
即，f(x)?k???x2?2?x???3.
??∵f(x)是可导函数，且x?1是它的唯一驻点，
∴x?1是f(x)的极值点，即有f(1)?k??1??2?1???3?2，解得k?2.
f(x)?2???x2∴?x???3?x2?2x?3. ?2??97.(1)解 当k??1时，原式 =
1b?(a?bx)kd(a?bx) ?1b?1k?1(a?bx)k?1?C； 当k??1时，原式 =
11b?a?bxd(a?bx)?1blna?bx?C.
(2)解
?3(1?2x)2dx??32?(1?2x)?2d(1?2x) ??312??2?1(1?2x)?2?1?C?32?11?2x?C. (3)解
?dx133?2x??12?(3?2x)?3d(3?2x)
112 ??12?(3?2x)?3?1?C??3(3?2x)3?C. ?143?1(4)13(x?1)x?1?13(x?1)x?1?C x31x211?x2(5)解 ?1?x2dx?2?1?x2d(x2)??12?1?x2d(1?x2)11?12?(1?x2)2(1?x2)?12?2?(1?x)2d(1?x2) 131?(1?x2)2?(1?x2)213?C?3(1?x2)3?1?x2?C.
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211(6)?e?2x?1?C (7)?sin(???x)?C
4?11(8)x?sin4x?C (9)2sinx?C 832(10)解
1?dx??1dx??x?x?d???tan?C.
11?cosx2cos2x2cos2x?2?22(11)解
?cosx?sinxd(sixn?coxs)11?2sinxcosxdx??(sixn?coxs)2??sinx?cosx?C. (12)2lnx?12(lnx)2?C
ln??1?1??ln???1?1??ln??1?1??(13)解 ??x?x(x?1)dx??x?x??1?x2??1?dx????1d?1??
?1?x??1??x?x2???ln???1?1?x??d???ln???1?1?x???????1?2??ln??1???1?x?????C.
(14)38x?14sin2x?132sin4x?C (15)sinx?13sin3x?C
(16)解 ?sin3xcos5xdx???sin2xcos5xd(coxs) ???(1?cos2x)cos5xd(cosx)???(cos5x?cos7x)d(cosx)
??16cos6x?18cos8x?C.
或 ?sin3xcos5xdx??sin3xcos4xd(sinx)??sin3x(1?sin2x)2d(sinx)
??(sin3x?2sin5x?sin7x)d(sinx)?1sin4x?1sin6143x?8sin8x?C. (17)?11115lncos5x?C(18)??C 3?sin3x?sinx(19)2sinx?2
5sin2x?sinx?C
(20)tanx?13tan3x?C (21)13tan3x?tanx?x?C
(22)13tan3x?C (23)?11?tanx?C
(24)12 6arctan312x?C (25)
22arctanx2?C
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(26)解
?dxdxx2?8x?25??(x?4)2?9?19?dx2
1???x?4??3???114?13??x?x?42d???1???x?4??3?3arctan3?C. ?3??(27)解
?dx?1dx4?x2arcsinx2?21???x?2
x?2??arcsi2n??1arcsind???arcsinx?2???lnarcsinx2?C. (28)解
?dxx?x2??dx12?2?dx??1?1?(2x?1)2
4??x?2????d(2x?1)1?(2x?1)2?arcsin(2x?1)?C.
(29)1122ln(x2?x?1)?3arctanx?13?C (30)1x?32lnx2?6x?13?4arctan2?C 85(31)18(1?x3)3?15(1?x3)3?C
lntanxx(32)解
?2lntanx2lntansinxdx??dx?dxsin?22xx2cos2tanxx22cos22??lntanx2d???lntanx?2???1?22??lntanx?2???C.
(33)13arcsin??3?2x????C
(34)解 ?11?ex?exex1?exdx??1?exdx??dx??1?exdx 43
?x??d(1?ex)1?ex?x?ln1?ex?C. (35)x?ln?1?ex??C (36)2ln?ex?1??x?C 98.(1)解 令1?x?t，则x?1?t，dx??dt.
原式 =??(1?t)21?2t?t2t100dt???t100????t?100?2t?99?t?98?dt ?199?11111t99?49?t98?97?t97?C ?11199?(1?x)99?49?1(1?x)98?197?1(1?x)97?C. (2)解 令x?t6，则dx?6t5dt.
原式 =?6t5?1?t3(1?t2)dt?6???1?1?t2??dt?6(t?arctant)?C ?6?6x?6arctan6x?C.
(3)解 令1?x?t，则x?t2?1，dx?2tdt. 原式 =?2t1?tdt?2????1?1?1?t??dt?2t?2ln1?t?C ?21?x?2ln(1?1?x)?C.
(4)解 令ex?1?t，则x?ln(t2?1)，dx?2tt2?1dt. 原式 =2?1t2?1dt????1?t?1?1?t?1??dt?lnt?1?lnt?1?C ?lnt?1ex?1?1t?1?C?lnex?1?1?C. (5)解 令x?sint，则dx?costdt. 原式 =?costcos3tdt??1cos2dt?tant?C 1 x ?x1?x2?C. 1?x2
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(6)a2xx2arcsina?2a2?x2?C (7)解 令x?atant，则dx?asec2tdt.
原式 =?sectdt?lnsect?tant?C
x2?a2 x ?lnx2?a2?x?C?lnx?x2?a2?C. a aa(8)
1x1a2?x2?a2?C (9)3ln3x?4?9x2?C (10)arccos1x?C(11)19?x2?9x?C
(12)
33ln3x?3x2?2?C (13)x2?a2?a?arccosax?C
99.(1)解
?te?2tdt??1?21??2t2?td(et)??2te??e?2tdt? ??12te?2t?14?e?2td(?2t)??12te?2t?14e?2t?C.
(2)(x2?2x?2)ex?C (3)2xsinxx2?4cos2?C
(4)14x2?14xsin2x?18cos2x?C (5)解 ?xsinxcosxdx?112?xsin2xdx??4?xd(co2sx) ??14(xcos2x??cos2xdx)??114xcos2x?8sin2x?C.
(6)xtanx?lncosx?x22?C (7)12secx?12lncscx?cotx?C (8)解 ?sec3xdx??secxd(taxn)?secxtanx??tan2xsecxdx
?secxtanx??(sec2x?1)secxdx?secxtanx??sec3xdx??secxdx?secxtanx??sec3xdx?lnsecx?tanx
移项，整理，得 ?sec3xdx?12?secxtanx?lnsecx?tanx??C.
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(9)xln2x?2xlnx?2x?C (10)解
?x2arctanxdx?13?arctanxd(x3) ?x33arctanx?13?x31?x2dx?x33arctanx?1?x?3???x?1?x2??dx ?x311d(1?3arctanx?x2)3?xdx?6?1?x2 ?x33arctanx?x26?16ln(1?x2)?C. (11)xarctanx?12ln(1?x2)?12(arctanx)2?C
(12)?217e?2x??xx??cos2?4sin2???C (13)解 ?sinlnxdx?xsinlnx??coslnxdx
?xsinlnx?(xcoslnx??sinlnxdx)
移项，整理，得 ?sinlnxdx?x2(sinlnx?coslnx)?C. (14)x(arcsinx)2?21?x2arcsinx?2x?C
(15)12(1?x2)ln(1?x2)?x22?C (16)13(x3?1)ln(1?x)?x39?x26?x3?C (17)?1x(lnx)2?2xlnx?2x?C (18)?cotx?lnsinx?cotx?x?C(19)?11?1xarcsinx?ln?x2x?C(x?1)
(20)2(x?2)ex?1?4arctanex?1?C
(21)exlnx?C (22)?e?xarctanex?x?1ln(1?e2x2)?C
(23)x2ex(sinx?cosx)?12excosx?C
100.解 ?xf?(x)dx??xd(f(x))?xf(x)??f(x)dx
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?xf(x)?sinx?C (1) x?xcosx?sinx?sinx?∵f(x)?? (2) ??2xx??∴将(2)代入(1)，得
?xf?(x)dx?x?xcosx?sinxsinx2sinx??C?cosx??C.
2xxx101.xlnx?C 102.(1)解 令
x?1A(x?1)(x?2)?x?1?Bx?2，得 x?1?(A?B)x?2A?B，
比较等式两边x同次幂的系数，得
??A?B?1?A???2A?B?1，解得??2?B?3. ∴原式= ?????2x?1?3?x?2??dx??2lnx?1?3lnx?2?C.
??x?1??(2)解 原式 =??1?4??dx， ?44x3?x????x?1x?1令 44x3?x?4x(2x?1)(2x?1)?Ax?B2x?1?C2x?1，
得
x4?1?(4A?2B?2C)x2?(B?C)x?A， 比较等式两边x同次幂的系数，得
????4A?2B?2CA?B?C?1?0?1 ，解得??7??B???A?14?8???C??98∴原式 =???1?4?1x?7191?8?2x?1?8?2x?1??dx
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?x4?lnx?78?12?d(2x?1)2x?1?98?12?d(2x?1)2x?1 ?x4?lnx?716ln2x?1?916ln2x?1?C. (3)13x3?32x2?9x?27lnx?3?C (4)x33?x22?x?8lnx?3lnx?1?4lnx?1?C (5)解 x2?1，x2?x?1都是二次质因式，因此令
1Ax?BCx?D(x2?1)(x2?x?1)?x2?1?x2?x?1， 得 1?(Ax?B)(x2?x?1)?(Cx?D)(x2?1)
?(A?C)x3?(A?B?D)x2?(A?B?C)x?B?D， 比较等式两边x同次幂的系数，得
??A?C?0?A????A?B?D?0?1C?0 ，解得?B?0?A?B?? . ??C?1?B?D?1??D?1∴原式 =???xx?1d(x2?1)1(2x?1)?1??1?x2?1?x2?x?1??dx??2?x2?1?2?x2?x?1dx??12ln(x2?1)?12?d(x2?x?1)1dxx2?x?1?2?x2?x?1
??12ln(x2?1)?12ln(x2?x?1)?12?dxx2?x?1.
∵
1dx1dx2dx2?x2?x?1?2???12??2 ?x??2???3341????2x?1??3????11?2x?1?123?2d?????arctanx?1?C， 1???2x?1??333???3???∴原式 =12ln??x?12x?1?1?x2?1???3arctan3?C. 48
(6)解 令 x2?1(x?1)2(x?1)?A(x?1)2?Bx?1?Cx?1， 得 x2?1?A(x?1)?B(x2?1)?C(x?1)2 ?(B?C)x2?(A?2C)x?A?B?C， 比较等式两边x同次幂的系数，得
??B?C?1?A??1?A?2C?0 ，解得??B?12 . ???A?B?C?1??C?12∴原式 =??????11111?(x?1)2?2?x?1?2?x?1???dx
?1x?1?12lnx?1?12lnx?1?C?1x?1?12lnx2?1?C.
103.(1)1?x (2)3x2sinx6 (3)
2x?x1?x8 (4)?xe
(5)解 f?(x)???0tx2?t??x3x2????x?tt3edt?0edt???????0edt??0edt??
??ex3?(x3)??ex2?(x2)???3x2ex3?2xex2. (6)解 f?(x)?2x??2x0e2tdt?x2?e4x?(2x)? ?2x?2x0e2tdt?2x2e4x.
(7)?extanln(e2x?1) (8)?sinx?coscosxcossin1?cos2x?cosx?x1?sin2x
104.解 在方程两边对x求导，得
f(x3?1)?3x2?1，f(x3?1)?13x2. 令x3?1?t，则x?3t?1，f(t)?13?f(x)?13(t?1)2，即3?(x?1).
32从而，f(1)?133?322?26. 49
x0105.(1) ∵limF(x)?0tf(t)dt0xf(x)x?0?limx?0x2?limx?02x?12lim1x?0f(x)?2f(0)
?0?F(0)， ∴F(x)在x?0处连续.
1x(2) ∵limF(x)?F(0)x2?x0tf(t)dtx?0x?limx?0?lim?0tf(t)dtxx?0x3
0?limxf(x)1f(x)00011x?03x2?3limx?0x?3limx?0f?(x)?3f?(0)，此极限存在.
∴F(x)在x?0处可导.
106.
cosxsinx?1
107.1?cos2(y?x)cos2(y?x)?2y 108.(1)1 (2)12 109.单调增加 110.在x?0处有极小值f(0)?0 111.最大值：f(0)?0；最小值：f(4)??323 112.f(x)?lnx?1 ????113.(1)解
?6660(2co2s??1)d???co2s?d(2?)??d???sin2??6?000?6?32??6. ???(2)解
?22220cosxsin2xdx?2?0cosxsinxdx??2?0cos2xd(coxs)
???2??1?2??3cos3x????20??0?1?23???3.
(3)
??43 (4)1 (5)ln32 (6)解
?1(ex?1)4exdx??10(ex?1)d(ex?1)?1x115(e?1)?5(e?1)500. (7)arctane??4 (8)32 (9)2 (10)(3?1)a (11)?2
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